Functional Stability Theory --- A Programmatic Hub: Universal Convexity-Uniqueness Across Scales (From Particles to Cosmology)

Functional Stability Theory (FST) is a research programme that proposes a single stability principle -- the Universal Convexity-Uniqueness criterion (UCU) -- as a unifying framework for emergent structure across physical and mathematical scales. This article is the programmatic hub of the FST series. It synthesises the FST-I (Thermodynamic Stability), FST-II (Chemical Stability), FST-III (Biological Stability) and FST-DE (Dark Energy) companion papers under a common architecture: universal game + MEPP axiom + UCU three-lemma endgame (UCU1 strict convexity, UCU2 existence, UCU3 quadratic lower bound) + RG bridge between scales. The paper is an honest survey: it tabulates which sub-problems are closed and which remain open (HD3' for Beal, full UCU3 for Yang-Mills, L4 RG-matching for the CRM-VI dark-energy line) and records the Pattern A methodological lesson learned from the cosmological route -- a positivity reformulation of an open conjecture is in general only a translation of language, not progress, unless its proof strength is independently verified. CHANGELOG Version 1.2 (May 2026) Structural clarification: FST-IV is the cosmological collector slot of the FST application series, consistent with the Plan.txt scale hierarchy (Level 1 = FST-I particles, 1b = FST-II chemistry, 1c = FST-III biology, Level 2 = FST-IV cosmology). FST-DE is positioned as a sub-component of FST-IV (specifically the dark-energy chapter of the cosmological collector). The collector slot itself is forthcoming as a separate paper and will integrate CRM I-IV, CRM-V (Saturation Theorem), CRM-VI (UV completion), and FST-DE under the FST-UCU framework. The series-glance table now shows the FST-IV collector slot explicitly, with FST-DE as a sub-component. The Hub paper (this document) retains its no-numeral position as the programmatic umbrella across all application scales -- not as a fourth numbered companion. No other content changes. Version 1.1 (May 2026) Title change: Former "Functional Stability Theory IV: ... -- An Overview" renamed to "Functional Stability Theory --- A Programmatic Hub: Universal Convexity-Uniqueness Across Scales (From Particles to Cosmology)". The Roman numeral "IV" was removed because the hub paper is positioned as a series umbrella, not as a chronological fourth instalment. Consistent naming with CORE/RFEP Physics-Master ("Functional Stability Theory --- Physical Instantiation via the Renormalized Free Energy Principle"). Bibliography update: The three FST-I/II/III companion bibitems now carry their Zenodo Concept-DOIs (FST-I 10.5281/zenodo.20130544, FST-II 10.5281/zenodo.20130563, FST-III 10.5281/zenodo.20130573). In v1.0 these were placeholders pending Phase-B reservation. Metadata: Related Identifiers now include all four hasPart relations (FST-I/II/III + FST-DE). Version stamp removed from title page (per LATEX_PDF_CHECKLISTE A2 -- Zenodo manages versioning). Version 1.0 (May 2026) Initial release as part of the FST series. Hub paper with cross-links to FST-I/II/III companions and references to the published FST mathematics and physics building blocks (RH, Zookeeper, Spectrum Duality, Hodge, BSD, P vs NP, Yang-Mills, Navier-Stokes, K41). Changelog: Structural clarification: FST-IV is the cosmological collector slot (consistent with Plan.txt scale hierarchy Level 1=I particles, 1b=II chemistry, 1c=III biology, Level 2=IV cosmology). FST-DE is positioned as sub-component of FST-IV. Series-glance table updated. The Hub paper retains its no-numeral position as the programmatic umbrella, not as a fourth numbered companion.